1. **State the problem:** We need to find the value of $e$ given that two lines intersect perpendicularly, forming four angles: one angle is $115^\circ$, another is $(5e)^\circ$, and the angle opposite to $(5e)^\circ$ is $e^\circ$. 2. **Understand the properties of intersecting lines:** When two lines intersect, opposite angles (vertical angles) are equal. 3. **Identify vertical angles:** The angle opposite to $(5e)^\circ$ is $e^\circ$, so these two angles are equal: $$e = 5e$$ 4. **Solve the equation:** $$e = 5e$$ Subtract $e$ from both sides: $$e - e = 5e - e$$ $$0 = 4e$$ Divide both sides by 4: $$\cancel{\frac{0}{4}} = \cancel{\frac{4e}{4}}$$ $$0 = e$$ 5. **Check the given angles:** Since $e = 0$, then $(5e)^\circ = 0^\circ$, which contradicts the presence of a $115^\circ$ angle and the fact that the lines are perpendicular. 6. **Use the fact that the lines are perpendicular:** The sum of adjacent angles formed by perpendicular lines is $90^\circ$. Given one angle is $115^\circ$, which is impossible for perpendicular lines (since adjacent angles must sum to $90^\circ$), the problem likely means the angles are supplementary or the $115^\circ$ angle is adjacent to $(5e)^\circ$. 7. **Assuming adjacent angles sum to $180^\circ$ (linear pair):** $$(5e)^\circ + 115^\circ = 180^\circ$$ Solve for $e$: $$5e = 180 - 115$$ $$5e = 65$$ Divide both sides by 5: $$\cancel{\frac{5e}{5}} = \cancel{\frac{65}{5}}$$ $$e = 13$$ 8. **Final answer:** $$\boxed{13}$$